Spectral functions and hadron spectroscopy in lattice QCD

Transcription

1 Spectral functions and hadron spectroscopy in lattice QCD Anthony Francis Jefferson Laboratory

2 Outline Biography Pre-academia Academia Spectral functions in lattice QCD at finite temperature - Transport and dissociation in heavy-ion collisions in the vacuum - HLO anomalous magnetic moment of the muon in multi hadron systems - Bound/Un-bound nature of the H-dibaryon Overview

3 Section 1 Biography

4 Biography: Pre-academia Born in Wrexham (Wales/UK) on and moved to Lemgo (Germany) in Completion of schooling by obtaining the Abitur in 06/2002 in Lemgo. Military service as medic with the German navy from 07/2002 until 04/2003. From 09/2003 position as Chef-du-Rang aboard QE2 of Cunard line until 04/2004. From 04/2004 Grundstudium of physics at Bielefeld University, Germany. Successful completion with the Vordiplom in 10/2005.

5 Biography: Academia Hauptstudium of physics at Bielefeld University, Germany, from 11/2005 to 10/2008. Thesis (12 months) in theoretical particle physics under the supervision of Prof. Dr. Edwin Laermann and Prof. Dr. Frithjof Karsch. Thesis project: Improved Staggered Lattice Meson Operators PhD in theoretical particle physics ( magna cum laude ) from 01/2009 to 10/2011 under supervision of Prof. Dr. Edwin Laermann, Dr. Olaf Kaczmarek and Prof. Dr. Frithjof Karsch at Bielefeld University, Germany. Thesis title: Thermal Dilepton Rates from Quenched Lattice QCD

6 Biography: Academia Post-Doc in Mainz since 11/2011 (later affiliated to the fairly new Helmholtz-Institut Mainz ). Close collaboration with Prof. Dr. Harvey Meyer and Prof. Dr. Hartmut Wittig. Further collaborations: Prof. Dr. Damir Becirevic at Universite Paris Sud XI, France, on Fermilab-type heavy quarks. Prof. Dr. Frithjof Karsch at Brookhaven National Laboratory, NY, USA, later also co-supervisor. Prof. Dr. Mikko Laine at Bielefeld University, now Albert Einstein Institute, Bern, Switzerland on diffusion from HQET.

7 Section 2 Spectral functions in lattice QCD

8 Spectral functions in lattice QCD Lattice QCD is formulated in Euclidean space-time For hadron spectroscopy the central observable on the lattice is the correlation function of two currents J µ : G µν (τ, T ) = d 3 x J µ (τ, x)j ν (0) (1) directly calculable in lattice QCD computations. However, there is also a different representation of the correlator as integral over the spectral function ρ(ω): G µν (τ, T ) = 0 dω 2π ρ cosh[ω(β/2 τ)] µν(ω, T ) sinh[ωβ/2] (2) only indirectly calculable via inverse Laplace-Transform. But, ρ(ω) is common both to Euclidean and Minkowski space-times. in this sense it is a more universal quantity.

9 Spectral functions in lattice QCD In the case of the electromagnetic current, assuming VMD, the spectral function ρ(ω) can be linked to via the simple relation: R(s) σ(e+ e hadrons) 4πα(s) 2 /(3s) (3) ρ(ω) ω 2 = R(ω2 ) 12π 2 where : s = ω 2 (4) F. Jegerlehner and A. Nyffeler, Phys.Rept. 477 (2009) 1110

10 Subsection 2 - Transport and dissociation in heavy-ion collisions

11 Spectral functions at finite temperature At finite temperature and especially in the deconfined phase the spf undergoes dramatic changes. Dissociation of bound state particles Emergence of transport phenomena / Transport phenomena Dissociation phen. T-> ( ) T=0 ( ) T>Tc ( )

12 Spectral functions at finite temperature These phenomena have visible effects in heavy-ion collisions Spf of light quarks Dilepton rates in the low energy regime Diffusion of heavy quarks Elliptic flow Dissociation of heavy quarkonium QGP thermometer Adare et al.; Phys.Rev. C84 (2011)

13 Charmonium spf via Maximum Entropy Method (MEM) MEM is a Bayesian technique that computes the most probable spectral function given some input model. Due to the gap between the transport and particle-peak regions, MEM works well here. Clear information for the diffusion coefficient and the dissociation pattern of the shown η c can be read off the spf. H.T.Ding, A.F., et al.; Phys.Rev. D86 (2012)

14 Heavy quark diffusion via lattice HQET In the quarkonium case the diffusion contribution can be isolated via the HQET correlator of the chromo-electric force: G E (τ) d 3 x J F (τ, x)j F (0) (5) lim M Note: No systematic extraction of the diffusion constant, yet. 4 tree-level improved T = 1.5T c, β = 6.872, 48 3 x 16 3 tree-level improved, pert. renorm., T ~ 1.43 T c T = 1.5T c, β = 7.192, 64 3 x 24 G imp / G norm T = 3.0T c, β = 7.457, 48 3 x 16 T = 3.0T c, β = 7.793, 64 3 x 24 Z pert G imp / G norm 2 1 NLO 16 x x x x τ T A.F., M. Laine, et al.; PoS LATTICE2011 (2011) τ T

15 σ el in the continuum limit of quenched QCD For light quarks here is no gap between the transport region and the continuous spectrum. MEM is inconclusive. Here: Eliminate lattice effects by taking continuum limit. Then fit σ el to physics constrained Ansatz. The fit result yields ρ(ω) and its parameters give σ el T 2 G ii ( T)/[ q G V free ( T)] 5 4 ii ( )/ T x x x x48 cont fit [0.2:0.5] T /T=0, /T=0 0 /T=1.5, /T=0.5 HTL free /T H.T.Ding, A.F., et al.; Phys.Rev. D83 (2011)

16 σ el in two-flavour QCD Much smaller lattices, continuum limit not feasible. But: Both T > T c and T 0 available. Exploit sum rule: 0 dω ω (ρ ii(ω, T ) ρ ii (ω, 0)) = Extract σ el from the intercept ρ(ω, T ). dω ρ(ω, T ) (6) ω B.B.Brandt, A.F., et al.; JHEP 1303 (2013) 100

17 Spectral functions at finite temperature Achievements so far Study of the electrical conductivity of light quarks in the continuum limit of quenched QCD. Study of heavy quark diffusion using HQET. Charmonium dissociation and diffusion via the Maximum Entropy Method. Extending the light quark study to dynamical ensembles and establishing the electrical conductivity also in this regime. Future goals Extend also the charmonium study to the dynamical regime. Take the continuum limit of the HQET inspired study. Increase the range of available temperatures in the dynamical regime. Study the dissociation of the ρ-particle accross the deconfinement phase transition.

18 Subsection 4 - HLO anomalous magnetic moment of the muon

19 Spectral functions and (g 2) µ The experimental observation and theoretical predictions of (g 2) µ show a dicrepancy of 3.4σ This computation is a precision test of the standard model. large non-perturbative uncertainties The leading hadronic order contribution to the anomalous magnetic moment of the muon constitutes one of the major hadronic uncertainties uncertainties (along with dominate light-by-light the overall scattering). uncertainty of the SM prediction a µ /10 11 Jegerlehner, Nyffeler, PR 477 (2009) Benayoun, Eur. Phys. J. C (2012) 72: QED incl 4-loops+LO 5-loops weak 2-loop lead. had. VP (experimentally e + e, τ) light-by-light (model) leading hadronic light-by-light scattering Table borrowed from A. Jüttner s presentation at Confinement X, 2012 in Munich, Germany.

20 Spectral functions, a HLO µ and ˆΠ(Q 2 ) We can write the leading hadronic contribution aµ HLO as ( α ) 2 aµ HLO = dq 2 K EW (Q 2, m µ ) ˆΠ(Q 2 ) (7) π where K EW (Q 2, m µ ) is a known electroweak kernel and ˆΠ(Q 2 ) = 4π 2 [Π(Q 2 ) Π(0)]. The key quantity to be calculated is therefore ˆΠ(Q 2 ), which can be written in terms of R(s) and consequently the spf ρ(ω) ˆΠ(Q 2 ) = Q2 3 0 R(s) ds s(s + Q 2 ) = 0 dω 2 4π2 Q 2 ρ(ω 2 ) ω 4 (ω 2 + Q 2 ) (8) Idea: Re-write ρ(ω 2 ) in terms of the correlator G(τ)!

21 A new representation of ˆΠ(Q 2 ) for lattice QCD Replacing ρ(ω 2 ) is indeed possible, the result is: ˆΠ(Q 2 ) = 0 dτ G(τ) [τ 2 4 Q 2 sin2 ( 1 Qτ)] (9) 2 This representation of ˆΠ(Q 2 ) is available at any value of the virtuality Q 2, while only the p = 0 correlator is required.... does not require an extrapolation of ˆΠ(Q 2 ) 0, eliminating one of the largest uncertainties in current lattice results.... comes at the cost of having to extrapolate the correlator to all times τ.... however, a Lüscher-type analysis and/or highly accurate spectroscopy poses a systematic route to reduce this cost.... in principle, also a highly accurate determination of the vacuum spf via e.g. MEM could render this issue irrelevant.

22 First results of the mixed-representation method Setup: lattice with m π = 324MeV and m π L = 5.0. Side remark: The mixed-rep. method also enables simply computing derivatives of ˆΠ(Q 2 ) by change of kernel. A.F. et al., Phys.Rev. D88 (2013)

23 Spectral functions and (g 2) µ Achievements so far Development of a new representation for ˆΠ(Q 2 ) in lattice QCD. Implementation and test of the new method. First results achieve a very good agreement with the standard method, without however having to extrapolate Π(0) or to use twisted-boundary conditions to boost the number of available virtualities. Future goals Repeat the analysis on all available CLS ensembles. Compute a HLO µ in the chiral and continuum limits. Develop strategy to fully control systematic uncertainties. Combine the two available representations to boost precision.

24 Subsection 6 - Bound/Un-bound nature of the H-dibaryon

25 Bound states in multi-baryon systems The study of multi-baryon systems poses a difficult challenge to lattice QCD and many interesting questions in this field are unanswered. One of these is the quark model prediction of a possibly stable six-quark state, the H-dibaryon (quark composition udsuds). Embarking on a study of the H-dibaryon, as the simplest multi-baryon system, some issues to be tackled are: The signal-to-noise ratio is expected to scale as the product of those of the individual baryons. The factorial growth of necessary quark contractions to form the desired system. We could handle part of these issues by using the newly developed, sophisticated algorithms put forward by the NPLQCD and HALQCD collaborations.

26 Bound states in multi-baryon systems To this extent we implemented a blocking -algorithm to carry out the necessary contractions In the next step we coded six different six-quark operators that all have overlap with the H-dibaryon in order to be able to set up a GEVP to compute the dibaryon-operator masses. X 1 Y 1 X 2 Y 2 = O X1 Y 1 (t)o X2 Y 2 (0) (10) where X i Y i ΛΛ; ΣΣ; NΞ;

27 First results of H-dibaryon masses Setup: lattice with m π = 451MeV and m π L = 4.7. The GEVP results are promising, However, they are not yet precise enough to decide on a bound or unbound nature of the H-dibaryon in our study ΛΛ ΛΛ GEVP 2 GEVP 1 GEVP 0 ma m Λ t/a A. F., C. Miao, T. D. Rae, H. Wittig; PoS LATTICE2013 (2013) 440

28 Bound states in multi-baryon systems Achievements so far Implementation of a blocking procedure to handle multi-baryon contractions. Set-up of the necessary code for studying the H-dibaryon. Spectrum analysis via GEVP. First results look promising Still large statistical errors. Future goals Increase statistics and reduce errors. Implement also non-relativistic operators. Go to larger, more chiral ensembles. In the farer future, go beyond the H-dibaryon and six-quark states.

29 Section 3 Overview

30 Overview el. conductivity (g-2) block-algo. H-dibaryon HQ diffusion scale setting Adler function multilevel-algo charmonium MEM MPI programming screening phen....etc...and much...more to come

31 Overview